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Final Exam Problems - Electromagnetic Fields II | PHYS 436, Exams of Physics

Material Type: Exam; Class: Electromagnetic Fields II; Subject: Physics; University: University of Illinois - Urbana-Champaign; Term: Fall 2007;

Typology: Exams

Pre 2010

Uploaded on 03/16/2009

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Physics 436 Final Exam December 15, 2007
3 hours. Open notes, closed book. Do all four problems: 10 points each.
1. Consider a plane wave in vacuum traveling along
ˆ
n=1
2
ˆ
x+ˆ
y
( )
, as shown:
a.
!
E
must be perpendicular to
!
k
. Which of Maxwell’s equations tells us this? Prove it.
b. Write an
!
E0
that describes linear polarization along the z-axis. The answer is not unique.
Write any
!
E0
that works.
c. Write an
!
E0
that describes a polarization in which the electric field rotates around
!
k
according to the right hand rule (i.e., right circular polarization).
d. For the
!
E0
in part c, write the corresponding
!
B0
.
2. Consider a material in which the relation between an electromagnetic wave’s k and
ω
is
k=a
!
, where a is a constant.
a. Calculate the phase and group velocities as functions of frequency.
b. Suppose we have a wave packet that would have a definite wavelength,
λ
, except that it is
limited to a finite length, L. Estimate the rate at which it spreads out. That is, roughly
what range of velocities is spanned by this wave packet?
(two more problems on the back)
y
ˆ
n
!
E=
!
E0ei
!
k!!
r"
#
t
( )
pf2

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Physics 436 Final Exam December 15, 2007

3 hours. Open notes, closed book. Do all four problems: 10 points each.

  1. Consider a plane wave in vacuum traveling along

n =

1

2

x +

y

, as shown:

a.

E must be perpendicular to

k. Which of Maxwell’s equations tells us this? Prove it.

b. Write an

E

0

that describes linear polarization along the z - axis. The answer is not unique.

Write any

E

0

that works.

c. Write an

E

0

that describes a polarization in which the electric field rotates around

k

according to the right hand rule ( i.e. , right circular polarization).

d. For the

E

0

in part c, write the corresponding

B

0

  1. Consider a material in which the relation between an electromagnetic wave’s k and ω is

k =

a

!

, where a is a constant.

a. Calculate the phase and group velocities as functions of frequency.

b. Suppose we have a wave packet that would have a definite wavelength, λ, except that it is

limited to a finite length, L. Estimate the rate at which it spreads out. That is, roughly

what range of velocities is spanned by this wave packet?

(two more problems on the back)

x

y

n

E =

E

0

e

i

!

k!

!

r " # t ( )

  1. Recall that an EM wave in a material has B

0

n

c

E

0

, where n is the index of refraction.

Because n is a property of the material, it is defined in the reference frame of the material.

Let’s consider a wave moving in the +

x direction that is linearly polarized along

y.

Suppose we observe the material to be moving along

x with velocity v (which may be

positive or negative).

a. What index of refraction, n! , do we measure for the propagation of the wave described

above?

b. Explain in words what happens when v =!

c

n

  1. To enhance the directionality of electric dipole radiation (wavelength λ), our intrepid

physicist places the source a distance d away from a conducting plane, as shown. The dipole

oscillates along z ˆ. The intensity distribution would be A sin

2

! in the absence of the

conductor. (I don’t want you to worry about all the constants.)

a. For a given λ, what value of d will maximize the intensity of radiation in the

y direction?

There are multiple solutions; I want the smallest one.

b. What is the angular distribution of the radiation intensity? Use spherical coordinates in

which z is the polar axis and f = 0 along

x.

z

d

p

x

y